Value at Risk (VaR): Calculation and Use

Value at Risk (VaR) summarizes the worst expected loss over a given horizon at a stated confidence level. A one-day 95% VaR of $10,000 means that, under normal market conditions, the portfolio is expected to lose no more than $10,000 on 95% of days — and to exceed that loss on the remaining 5%.

The three calculation methods

Historical VaR uses the empirical distribution of past returns. Sort historical daily returns and read the loss at the chosen percentile. It makes no distributional assumption but assumes the past is representative of the future.

Parametric (variance-covariance) VaR assumes returns are normally distributed. For a portfolio with standard deviation $\sigma$ over horizon $h$:

$$\text{VaR}{\alpha} = -(\mu_h - z{\alpha} \cdot \sigma_h)$$

where $z_{\alpha}$ is the standard normal quantile (1.645 for 95%, 2.326 for 99%). For a one-day horizon, $\mu_h$ is usually set to zero for conservatism.

Monte Carlo VaR simulates thousands of return paths from an assumed distribution, then reads the percentile. More flexible than parametric, but inherits all assumptions of the chosen model.

Scaling across horizons

Under the square-root-of-time rule, VaR scales as:

$$\text{VaR}_h = \text{VaR}_1 \cdot \sqrt{h}$$

This holds only if returns are independent and identically distributed. With autocorrelation, volatility clustering, or mean reversion, the rule under- or over-states longer-horizon risk.

Practical use for traders

• Position limits: Set VaR limits per position, strategy, and book. A breach triggers a reduction in size.
• Capital allocation: Allocate risk budget in VaR terms rather than capital. Two strategies with equal capital but different VaR carry very different risk.
• Stress overlay: Use VaR for normal conditions and supplement with stress scenarios for tail risk.

The known weaknesses

• Not a measure of tail loss. It tells you the threshold, not what happens beyond it. Two portfolios with identical 95% VaR can have very different losses in the worst 5%.
• Assumes the recent past predicts the future. Historical VaR failed in 2008 because the prior year contained no comparable stress.
• Hides fat tails. A 99% VaR of $5,000 says nothing about whether the 99.5% loss is $6,000 or $60,000.
• Breeds false precision. Reporting VaR to the dollar implies accuracy the data does not support.

Using VaR sensibly

Pair VaR with expected shortfall to capture tail behavior. Stress-test the assumptions by re-estimating VaR after removing the worst historical day — if the number jumps dramatically, the portfolio is fragile. VaR is a useful summary statistic, not a risk management system. Never let it substitute for stress testing and judgment about what could break that the model has never seen.